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A forward-backward dynamical approach for nonsmooth problems with block structure coupled by a smooth function

In this paper we aim to minimize the sum of two nonsmooth (possibly also nonconvex) functions in separate variables connected by a smooth coupling function. To tackle this problem we chose a continuous forward-backward approach and introduce a dynamical system which is formulated by means of the partial gradients of the smooth coupling function and the proximal point operator of the two nonsmooth functions. Moreover, we consider variable rates of implicitness of the resulting system. We discuss the existence and uniqueness of a solution and carry out the asymptotic analysis of its convergence behaviour to a critical point of the optimization problem, when a regularization of the objective function fulfills the Kurdyka-Lojasiewicz property. We further provide convergence rates for the solution trajectory in terms of the Lojasiewicz exponent. We conclude this work with numerical simulations which confirm and validate the analytical results.